Reliability analysis of generating systems including intermittent sources

Reliability analysis of generating systems including intermittent sources S Fockens, A J M van Wijk and W C Turkenburg Department of Science, Technology and Society, University of Utrecht, The Netherlands

C Singh Department of Electrical Engineering, Texas A&M University, College Station, TX 77843. USA A new method for the calculation of expected unserved energy of large-scale power systems including intermittent energy sources like hydro power units, wind turbines and photovoltaic units is presented. The correlation between hourly load and intermittent energy supply is accurately accounted for by the hourly application of a mean capacity outage table. The proposed method is efficient, because it eliminates the need for hourly computation of a system negative margin table. In addition, expressions are given for computing the loss of load expectation and loss of load frequency. The method is applied to the IEEE reliability test system which is extended with intermittent sources in the form of wind turbine units. Keywords : reliability scheduling, generation system reliability, alternative energy sources

I. I n t r o d u c t i o n Technical development of renewable energy sources, like wind turbines and photovoltaic units, has reached the stage where these sources are being included in large-scale generating systems. Growing recognition of environmental problems will further increase the contribution of wind and solar energy to future generating systems. Because of their intermittent nature the contribution of unconventional sources to the total system reliability has a time-dependent character. For planning purposes it is important to know what contribution the unconventional sources make to the total system reliability. Generally, three basic indices are used to indicate generating system reliability. The loss of load expectation (LOLE) denotes the expected number of hours per year during which the system load exceeds the available generating capacity. The expected unserved energy .(EUE) is defined as the expected amount of energy not supplied by the generating system during the period of observation, due to capacity deficiency. The loss of load frequency ( L O L F ) is the expected number of times Received 1 June 1 991 ; accepted 1 January 1 992

2

that a situation of capacity deficiency occurs in this period. If a reliability analysis method is to include unconventional sources, it should take into account the following factors1 : scheduled outage, failure and repair of both conventional and unconventional units, the fluctuating nature of the available unconventional energy supply, and the correlation between the fluctuating energy supply and the hourly demand. A concise and accurate method for the calculation of L O L E and L O L F has been described 1. It is realized that future generating systems may be energy-limited rather than capacity-limited when intermittent sources come to be used more extensively, so EUE is a more appropriate reliability measure than LOLE. Unfortunately, the method described in Reference 1 is considered to be unsuitable for EUE calculation since computation time will increase drastically 2. This increase is because one needs to construct a table with negative margin states for the total system in order to calculate the hourly contribution to EUE. This requires an elaborate hourly convolution of the outage tables of the subsystems. To reduce the number of times that this timeconsuming hourly convolution needs to be performed, a clustering technique has been proposed 2 for the calculation of EUE and LOLE. The original set of hourly data is replaced by a set of clusters. Because the number of hours is reduced to a relatively small number of clusters, the computation time is considerably reduced. However, the approach proposed in this paper has the following advantages over the clustering approach •

The proposed approach is exact for the given data, whereas clustering involves a certain amount of approximation. The load values in the clusters are averages over a number of hours that are correlated and this averaging is the source of approximation. Although reasonably accurate results can be obtained if one uses a sufficiently large number of clusters and makes a suitable choice of initial seeds in the clustering process, the indices computed with clustering are not exact.

0142-061 5/92/010002-07 © 1992 Butterworth-Heinemann Ltd

Electrical Power & Energy Systems

•

•

The proposed method can yield the contribution to EUE and L O L E for individual hours whereas the clustering approach cannot. This information might be needed if one has to detect the hour with the largest loss of load probability. As described later in the text, the proposed method can be applied to simultaneous computation of EUE, L O L E and L O L F in an efficient manner, whereas the clustering approach is only suitable for the computation of EUE and LOLE.

In this paper we start by describing the system model used, then we briefly explain how EUE is computed by means of negative margin probabilities in the clustering approach ~. We demonstrate how the concept of mean capacity outage can be used to simplify EUE calculation on an hourly basis. The proposed method is illustrated by means of a small sample-system. In addition, we give expressions for calculating L O L E and LOLF, so that all three indices can be calculated simultaneously. Finally, the proposed method is applied to the IEEE reliability test system which is extended with wind turbine units and we compare the efficiency of the proposed approach with the clustering approach.

II. System model The system load is described as a chronological sequence of N t discrete load values L i for successive time steps i = 1,2, 3 , . . . , N r Each time step has equal duration AT = T/Nt, where T represents the total duration of the observation period. The generating system is divided into a subsystem containing all the conventional generating units and a separate subsystem for each type of unconventional unit, e.g. wind turbines and photovoltaic units. For simplicity of presentation we assume initially that the generating system contains only one unconventional subsystem. We use subscripts c and u to refer to the conventional and unconventional subsystems, respectively. Later we will extend this notation to an arbitrary number of unconventional subsystems. From the reliability characteristics of the conventional generating units, a system generation model can be constructed, using the well known concept of recursive unit addition 3. This model consists of Nc discrete capacity outage levels, Xc(k), where k = 1, 2, 3 . . . . . N~, which are arranged in strictly ascending order, Xc(k + 1 ) > X~(k ). Note that Xc ( 1 ) = 0 and X (N c) = Cc, where Cc represents the total rated capacity of the conventional subsystem. The notation Pc(k) and lee(k) is used to denote the cumulative probability and frequency 3 with which outages occur, X c > X c(k). The incremental probability Pc(k) is defined4'5 as the probability that an outage occurs in the conventional subsystem Xc = X~(k). Note that the cumulative probability can be expressed in terms of the incremental probabilities as (1)

r=k

Similar to the capacity model of the conventional subsystem, {Xc(k), Pc(k),ffc(k)}, a separate capacity model of the unconventional subsystem is constructed: {X,(j),/~u (J), if, U)} where j = 1, 2, 3 . . . . . N.. The effect of fluctuating energy in the unconventional subsystem is included by a factor Ag, 0 -%
Vol 14 No 1 February 1992

Nt

EUE = AT ~ Ui

(2)

i=1

where Ui represents the expected unserved load during the time step i U,=

~

(3)

(X-X,)P(X)

X>Xi

For each hour i the critical capacity outage is defined as

(4)

X, = Cc + A,C.. - L,

where Cc + A~C, represents the effective total system generating capacity during hour i in the case where all units are available. Capacity deficiency occurs if the total system outage X exceeds X~.

III. Negative margin method For easy comparison with the proposed method, we briefly describe the negative margin method for EUE calculation in generating systems that include unconventional units t . The margin or capacity reserve is defined as the excess of available capacity over load demand M = (Cc + A , C . -

X ) - L, = X , -

X

(5)

A negative margin occurs when the load exceeds the available capacity. Let P,,(M) represent the probability that a margin M occurs. Now equation (3) can be rewritten as U,=

~

fMIP,.tM)

(6)

M<0

However, computation of Pro(M) for all individual margins would require considerable computational effort in the case of a realistic system with many outage states. To reduce computation time an approximate expression is derived 2'7

U, = AM

(0) +

~, Pm(M)

(7)

M= -AM

where P,. (M) represents the cumulative probability that a margin occurs equal to or less than M N=

Pro(M) = ~, .°c(hu)P.(j)

Nc

/~(k) = ~ Pc(K)

fraction of the total rated unconventional power that is effectively being generated during hour i. For each hour i the capacity outage vector X.(j) is replaced by a modified vector X.i(j) = AiX.j, while the vector P, (j) remains unchanged. In addition, every hour the capacity models of the two subsystems can be combined or convolved in order to compute the capacity model of the total system: { X , P ( X ) } , where the notation P ( X ) is used to represent the probability that a system capacity outage occurs exactly equal to X MW. A general expression L6 for the computation of the expected unserved energy is

(8)

j=l

For given values of i and j, the index hu is defined as the smallest integer which satisfies X, - X = X, - X~(hu) - X.,(.j) <~ M

(9)

In the summation of equation (7) the cumulative probability P,.(M) is calculated at discrete negative margins M = - AM, - 2AM, - 3AM . . . . . - L ~ where

3

Li is the smallest possible negative margin during hour i and where AM is a fixed positive increment value. The accuracy of U~ computed with equation (7) depends on the choice of the margin increment AM. -

IV. Proposed method In this section we will demonstrate how the computation of the system negative margin table avoided by the application of a mean capacity table 6 for the computation of U/. Let P/ represent the loss of load probability hour i P,=

hourly can be outage during (10)

~, P ( X ) X>Xi

and let Hi represent the expected value or mean value of all capacity outages which would cause capacity deficiency during hour i X>Xi

Using equations (10) and (11 ) we can rewrite equation (3) as (12)

U~ = Hi - X,P/

Explicit evaluation of (10) and ( 11 ) with summation over all outage states, X = Xc(k) + X.i(j) > X~, for relevant combinations o f j and k yields, respectively Nc

P/= ~ ~ Pc(k)P.(j) j=l

(13)

N~

k=kij

Hi = ~

X,,(j)]Pc(k)P,(j)

(14)

Pi

(20)

/=1

Using Reference 1 we present expressions for LOLF using the notation of this paper AT N, LOLF = - - ~ (F~/+ F~ + FT) T

(21)

/=1

where F~ and F~' are components of frequency due to interstate transitions in the conventional and unconventional subsystems, whereas F I is the frequency component due to load variation and variation in the unconventional energy supply. These components are given by

where k u is defined such that Xc(ku) represents the smallest conventional capacity outage that would cause capacity deficiency for given values of i and j, i.e. kq is the smallest integer which satisfies

X~(k/~) + X.,(j) > X~

(22)

F~ = ~ [Pc(k,j)- Pc(k,.j_l)]F,(j)

(23)

j=2

Fli = { ~i+I - PI)/AT ifPi+l > ~ Pi

(24)

(15)

Substitution of equation (1), with k = kq, in equation (13) yields a simplified expression for the loss of load probability N.

P~ = ~ Pc(k/j)P.(j)

(16)

j=l

A similar substitution in equation (14) allows us to rewrite the mean capacity outage during hour i as Hi = ~ P,(J') X./(j)/3c(k/~)+ ~ j= 1

F~ = ~ Fc(kq)P.(l) j=l

j= 1 k=kij

Xc(k)Pc(k)

(17)

k=kij

In order to simplify equation (17) we define N~

Be(k) = ~ xc(,c)P~(~c)

(18)

r=k

Substitution of equation (18), with k = ko, in equation ( 17 ) yields Nu

Hi = ~, [/4c(k u) + X,i(j)-Pc(kifl]P.(j)

(19)

j=l

We refer to/qc(k) for k = 1, 2, 3 . . . . . N~, as the mean capacity outage table of the conventional subsystem. This table is the key concept of the proposed method for EUE computation. The general concept of a mean capacity

4

T

N~

Nc

~, [Xc(k) +

AT N, LOLE . . . . . . ~

(II)

Hi= ~ XP(X)

N~

outage table has been introduced elsewhere 6. After the cumulative outage probability table Pc(k) has been constructed, construction of the mean capacity outage table -Oc(k) requires little additional computational effort because one can use a simple recurrence relation 6. Comparison of equations (13) and (14) with (16) and (19) shows that, through application of the cumulative outage probability table Pc(k) and the mean capacity outage table/~c (k), we have been able to replace the double summation over j and k by a single summation over j. In the negative margin method the summation over j in equation (8) needs to be performed for each negative margin, whereas in the proposed method the summation is performed only once in equations (16) and (19). The definition of P/ in equation (10) is chosen such that the loss of load expectation can be expressed as

Note that the computation of F~ and F~' by means of equations (22) and (23) involves summations over j which are similar to those involved in the computation of P/ and H/ using equations (16) and (19). Thus, simultaneous computation of EUE, LOLE and LOLF can be performed efficiently on an hourly basis since computation of P/, Hi, F~ and F~' can be combined in one single loop over j, so that for each j the value of k/j need only be determined once. In summary, the proposed method for EUE, LOLE and LOLF computation proceeds in the following steps (1) Construction of the capacity outage table Xc(k), the cumulative outage probability and frequency tables Pc(k) and Fc(k ) for the conventional subsystem, using the unit addition algorithm 3'7. (2) Construction of the capacity outage table X, (j), the incremental outage probability table P.(j) and the cumulative outage frequency table F.(j) for the unconventional subsystem, using the unit addition algorithm. (3) Construction of the mean capacity outage table /~c(k) for the conventional subsystem, using the recurrence approach 6. (4) Modification of the outage vector X.(j) to obtain the vector X.i(j) = AiX.(j) for each hour i.

Electrical Power & Energy Systems

(5) Computation of the hourly contributions to the indices using equations (12), (16), (19) and (22)-(24). (6) Computation of EUE, LOLE and LOLF by summation over all hours using equations (2), (20) and (21). Note that all tables are only constructed once, at the beginning of the observation period, except for the capacity outage vector X,i(j), which is modified each hour. Computation of X,~(j) is simple and fast since it involves only multiplication of X, (j) by Ai. In practical applications, when planned outage is considered on a weekly basis, all tables constructed in steps 1, 2 and 3 must be reconstructed or modified at the beginning of each week 6. IV.1

Sample-systemanalysis

To illustrate the proposed method we compute EUE for a small generating system with total rated capacity 33 MW. The conventional subsystem contains 3 identical 2-state generating units each with 10 MW capacity, failure rate 0.1 h -1 and repair rate 0.9h -1. The unconventional subsystem contains 3 identical 2-state units each with 1 MW capacity, failure rate 0.1 h -~ and repair rate 0.4 h- 1. For simplicity, we consider a single hour i with system load L~ = 20.4 MW and derating

factor of the unconventional capacity Ai = 0.8, so that the modified capacity outages of the unconventional subsystem are X,i(j) = 0.0, 0.8, 1.6 and 2.4 MW. For both subsystems the incremental, cumulative and mean capacity outage tables are given in Tables 1 and 2, respectively. The critical capacity outage is calculated by applying equation (4). With conventional capacity Cc = 30 MW and unconventional capacity C, = 3 MW, we find X~ = 30 + 0.8 x 3 - 20.4 = 12.0 MW. Table 3 gives the relevant part of the total system capacity model during the hour i. The smallest capacity outage which would cause capacity deficiency during hour i is 12.4 MW. Explicit evaluation of equation (3) with substitution of incremental probabilities taken from Table 3, P(X), X = 12.4, 20.0 . . . . . 32.4MW, yields U~ = 0.2482176 MW. Computation of U~ by the application of equation (7) and summation over all negative margins with AM = 1 MW yields U~ = 0.237136 MW. This result differs slightly from the value found using equation (3) because (7) is a numerical approximation of Ui. Now we will demonstrate how U~ can be calculated with the proposed expressions (12), (16) and (19) without explicit construction of a negative margin table. Application of equations (16) and (19), with k u = 3 for j = 1 , 2 , 3 and ku = 2 for j = 4 , and substitution of

Table 1. ConVentional subsystem capacity model Outage index

Capacity outage

Increm. prob.

k

Xc(k) (MW)

Pc(k)

Cumul. prob. /~c(k) (MW)

Mean outage He(k) (MW)

Cumul. freq. /ec(k ) ( h - ' )

1 2 3 4

0.0 10.0 20.0 30.0

0.729 0.243 0.027 0.001

1.000 0.271 0.028 0.001

3.00 3.00 0.57 0.03

0.0000 0.2187 0.0486 0.0027

Table 2. Unconventional subsystem capacity model Outage index j

Capacity outage X=(j) (MW)

Increm. prob. P=(j)

Cumul. prob. P=(j)

Mean outage H=(j) (MW)

Cumul. freq. F=(j) (h -~)

1 2 3 4

0.0 1.0 2.0 3.0

0.512 0.384 0.096 0.008

1.000 0.488 0.104 0.008

0.600 0.600 0.216 0.024

0.0000 0.1536 0.0768 0.0096

Table 3. System capacity model Capacity outage X (MW)

Increm. prob.

P(X)

Cumul. prob. P(X)

Mean cap. outage / t ( X ) (MW)

Cumul. freq. /~(X) ( h - ' )

12.4 20.0 20.8 21.6 22.4 30.0 30.8 31.6 32.4

0.001944 0.013824 0.010368 0.002592 0.000216 0.000512 0.000384 0.000096 0.000008

0.029944 0.028000 0.014176 0.003808 0.001216 0.001000 0.000488 0.000104 0.000008

0.6075456 0.5834400 0.3069600 0.0913056 0.0353184 0.0304800 0.0151200 0.0032928 0.0002592

0.0522936 0.0486000 0.0292464 0.0095472 0.0033264 0.0027000 0.0014712 0.0003576 0.0000312

Vol 14 No 1 February 1992

5

V. System case study

values taken from Tables 1 and 2 yields P1 = 0.029944 and Hi = 0.6075456 MW. By application of equation (12) we find Ui = 0 . 6 0 7 5 4 5 6 - 12.0 x 0.029944 = 0.2482176 MW, which is identical to the value previously found by explicit evaluation of equation (3).

To evaluate the proposed method we used the IEEE reliability test system (RTS) described in Reference 8. The conventional generating system contains 32 binary units with unit capacities ranging from 12 to 400 MW and the system capacity is 3405 MW. In this study one unconventional subsystem is added to the RTS. This subsystem consists of identical wind turbines each with an installed capacity of I MW, a mean up time of 190 h and a mean down time of 10 h. Typical hourly mean wind velocity data are used for The Hook of Holland, The Netherlands 9. The velocities are converted into effective hourly capacities, which are given in Table 4. To obtain the hourly derating factors the values in Table 4 must be divided by 1000MW. Four cases are considered: a base case with no unconventional capacity and three cases with C, = 100, 200 and 400 MW, respectively. In all four cases the conventional system is the complete RTS without modification. Reliability indices are calculated for a period of one week. Hourly load values are obtained 8, using the load cycle for week 51 with peak load 2850 MW, low load 1368MW and weekly energy demand 359.3GWh. Numerical results for EUE, LOLE and L O L F computed with the proposed method are given in Table 5. For an increasing amount of unconventional capacity the value of all three indices decreases. In the last column of Table 5 the CPU-times are given for the computation of all three indices on an IBM-AT personal computer. The CPU-time required (11.7 s) for the construction of the RTS capacity model is indicated separately, For comparison, we also computed EUE using equations (7) and (8) with a fixed margin increment AM = 10 MW and clustering with the nearest centroid sortin9 algorithm as described in Reference 2. The accuracy with which EUE is computed depends on the number of clusters, the choice of initial seeds in the clustering algorithm, and the correlation between hourly load and wind energy supply. For several numbers of clusters, N o = 10, 15, 20, 40 and 60, the value of EUE is computed and presented in Table 6. As initial seeds we used the N o hours with the highest load. For N o = 15 we find a deviation of 15.0%, 15.6% and 13.8% from the exact value of EUE for C, = 100, 200 and 400 MW, respectively. In Figure 1 we have plotted the quotient of EUE computed with clustering and EUE computed with the proposed method for C, = 400 MW as a function of the number of clusters used. Note that the values of EUE with clustering are systematically lower than EUE computed with the proposed method because the impact of the peak values

IV.2 Multiple unconventional subsystems So far we have assumed, for simplicity of presentation, that the system contains only one unconventional subsystem. However, in practice the system might include several unconventional subsystems: one subsystem for each type of intermittent source like hydro power plants, wind turbines or photovoltaic units. In addition, one type of intermittent source groups of units might form separate subsystems because the hourly energy production for each group is characterized by a separate time series Ai. For example, different types of wind turbines each have specific power curves 9. In this subsection we present expressions for the computation of Pi and Hi where more than one conventional subsystem is included in the system. Let n represent the number of unconventional subsystems. We use the subscript u = 1,2, 3 . . . . , n to index the unconventional subsystems. The notations X, (j,) and P, (j,) refers to the capacity outage level and the capacity outage probability of subsystem u, respectively. To simplify the notation, we use the shorthand j =--Ja,J2,J3 . . . . . j,, and

P(J)- f i P.0'.)

(25)

u=l

X(i,j)-

~ A.,X.(j.)

(26)

u=l

where A,i represents the fraction of the total rated power of subsystem u that is effectively being generated during hour i due to fluctuating energy. Instead of equations (4), (16) and (19) we use, respectively XI=C~-L~+

~ A.~C,

(27)

U=I

P, = ~ Pc(ku)P(j )

(28)

i

H, = ~ [/~c(k,j) + X(i, j)P,(k,j)]P(j)

(29)

J

where k u is the smallest integer which satisfies

X~(kij) + X(i,j) > Xl

(30)

The summation over j in equations (28) and (29) is over all combinations of indices jl, J2,J3 . . . . . Jn which satisfy

Cc + X ( i , j ) > X,

(31)

Table 4. Effective hourly power output (MW) of 1000 MW installed wind turbine capacity Hour Day

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

1 2 3 4 5 6 7

150 439 378 326 379 745 649

85 197 379 276 378 565 565

150 290 379 565 378 565 493

150 131 379 493 369 548 639

192 267 276 265 325 576 548

240 305 81 240 239 576 680

345 436 120 171 459 821 738

293 255 34 115 447 429 912

425 69 69 167 153 423 870

420 102 24 295 270 205 865

572 103 24 534 590 699 695

623 189 24 829 270 497 697

539 189 221 752 709 907 956

640 379 153 709 957 909 938

749 545 163 878 947 957 656

570 540 289 824 949 954 742

489 565 487 850 939 313 939

547 579 305 786 812 912 876

800 580 323 726 871 879 937

716 436 234 493 766 933 914

480 339 234 493 766 951 831

319 321 326 565 649 874 766

425 378 433 493 649 745 916

439 378 379 379 565 874 575

6

Electrical Power & Energy Systems

Table 5. Case results for proposed method

Table 7. CPU time comparison for EUE computation

C, EUE (MW) (MWh)

LOLE (h/week)

LOLF C P U time (occ./week) (s)

N.

No

AM

t 1 (S)

t 2 (S)

t2/t 1

0 100 200 400

1.951174 1.487951 1.185692 0.789840

0.387078 0.310602 0.258305 0.193275

100 200 400 100 200 400

20 20 20 60 60 60

90 86 78 103 99 89

11.5 23.3 45.2 11.5 23.3 45.2

100 190 341 350 660 1185

8.7 8.2 7.5 30.4 28.3 26.2

278.917 207.902 159.402 99.085

11.7 11.7 11.7 11.7

+ + + +

0.4 18.3 36.5 72.5

IMol/ 8.7 8.3 7.5 29.8 28.6 25.7

Table 6. EUE computed with clustering (MWh) Numberof clusters

equations (16) and (19). The EUE computation time using both clustering and explicit evaluation of negative margins can be approximated by

c. (MW)

10

15

20

40

60

80

100 200 400

155.6 117.5 73.0

176.8 134.5 85.4

185.5 143.1 90.0

198.7 153.2 96.1

206.0 158.0 98.2

207.7 159.2 98.9

°a:noco 0.9

0 0

O0

W

0.8 o

0

W

7-

0

O0

hi

0

0.7 ,

0

I

,

IO

I

,

20 Number

I

I

50

40

of clusters

(33)

where ~2 represents the average CPU-time required for the evaluation of a single term in the summation of equation (8). The fraction IMoI/AM represents the average number of negative margins that is evaluated in equation (7) per cluster. Note that in equations (32) and (33) the time required for the construction of the capacity models is not included. The efficiency improvement of the proposed method over the existing method can be expressed as

1.0

W

t2 = ~2N, Ng IMol AM

,

I

,

t2 tl

-

~2 No IMol

(34)

~1 N AM

For No = 40, N t = 168, N, = 100 and AM = 10 MW we find Mo = - 1010 MW, t 1 = 11.5 s and t 2 225 s. By substituting tl and t 2 into equations (32) and (33), respectively, we estimate ~1 = 0 . 6 8 5 m s and ~2 = 0.557 ms, so ~t2/~ I = 0.81 for our computer implementation. To verify equation (34), we determined Mo and measured both tx and t 2 for No = 20 and No = 60. The results are presented in Table 7: the fractions t 2 / t 1 and the values of 7 computed with equation (34) diverge by less than 2%. =

50

60

Ng

Figure 1. EUE computed with clustering versus NQ

is reduced in the process of obtaining average values. For a sufficiently large number of clusters, Ng/> 60 in this case, the error in EUE computed with clustering becomes less than 1%.

VII. Discussion VI. C o m p u t a t i o n

t i m e analysis

In general it is difficult to compare the computation time required for different algorithms. In this section we attempt to express the time required for the computation of EUE as a function of some characteristic parameters. For all computations with explicit evaluation of negative margins we truncated the summation in equation (7) at a cumulative probability of 10 -8, i.e. margin probabilities P r o ( M ) < 10 -8 are not computed since they do not contribute significantly to the value of Ui. Let Mc represent the smallest negative margins for which P,,(M) is computed, averaged over all hours i. The CPU-time required for EUE computation by application of the proposed method can be approximated by tl = ~ N . N t

(32)

where ~ is the average CPU-time required for the computation of a single term in the summation of

Vol 14 No 1 February 1992

The method described in this paper yields exact results for the given data. However, in practice, the time series representing the hourly energy production of unconventional sources will be different from one period to another. This means that the choice of the data series determines the actual value of the computed indices. Therefore, time series of several time periods must be used if realistic values of the reliability indices are to be obtained. Another point of consideration is the magnitude of the time step used. In the case of wind turbine units, fluctuations of wind power within the hour can be significant, so a time step of one hour with average wind power might not be adequate.

VIII. C o n c l u s i o n s The proposed method is able to compute the basic reliability indices EUE, L O L E and L O L F in generating systems including time-dependent sources. The existing method for L O L E and L O L F computation in such

7

systems 1, which is considered to be unsuitable for EUE computation 2, is extended to include computation of EUE with little additional computational effort. This offers a better alternative than clustering 2, although the proposed method can also be applied in combination with clustering provided an adequate number of clusters are used. The proposed method is conceptually simple, accurate and more efficient than the clustering approach.

IX. Acknowledgments The authors would like to thank S M McNab for her linguistic assistance. The research described was supported financially by the Dutch Ministry of Education and Sciences (O&W).

X. References 1 Singh, C and Lago-Gonzalez, A 'Reliability modeling of generation systems including unconventional energy sources' /EEE Trans. Power Appar. & Syst. Vol PAS-104 No 5 (1985) pp 1049-1056

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2 Singh, C and Kim, Y 'An efficient technique for reliability analysis of power systems including time dependent sources' IEEE TransPowerSyst Vol PWR-3 No 3 ( 1988 ) pp 1090-1096 3 Billinton, R and Allan, R N Reliability Evaluation of Power Systems Pitman, London (1984) 4 Wang, X and Pottle, C "A concise frequency and duration approach to generating system reliability studies' IEEE Trans PowerAppar&SystVol PAS-102 No8 (1983) pp2521 2530 5 Wee, C L and Billinton, R "A frequency and duration method for looped configuration generating capacity reliability evaluation' IEEE Trans Power Syst Vol 3 No 2 (1988) pp 698-705 6 Fockens, S, van Wijk, A J M, Rurkenburg, W C and Singh, C 'A concise method for calculating expected unserved energy in generating system reliability analysis"/EEE Trans Power Syst Vol PWRS-6, No 3 (1991) pp 1085-1091 7 EPRI Technical Report EL-2519, Vol 1 "Modelling of unit operating considerations in generating capacity reliability evaluation' Final Report, EPRI, July 1982 (Research Project 1534-1 ) 8 Report by the Reliability Test System Task Force of the Application of Probability Methods Subcommittee 'IEEE Reliability Test System" IEEE Trans Power Appar & Syst Vol PAS-98 No 6 (1979) pp 2047-2054 9 van Wijk, A Wind energy and electricity production PhD Thesis University of Utrecht, The Netherlands, June 1990

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